1
00:00:00,450 --> 00:00:10,979
Our starting point for the logistic equation is the simple model that I discussed in the last video, f of p equals rP.
2
00:00:11,090 --> 00:00:15,684
This has the rather unrealistic feature that populations grow without bound. ¶
3
00:00:15,958 --> 00:00:20,795
The world would be completely taken over by rabbits in the previous example.¶
4
00:00:21,189 --> 00:00:30,232
We know that's not the case, because we can move about the world without hitting rabbits, so we know in general that populations don't grow forever.
5
00:00:30,386 --> 00:00:37,055
So we'd like to modify this in some simple way, to account for the fact that there's going to be some limit to the number of rabbits,
6
00:00:37,065 --> 00:00:40,112
or whatever it is we're studying.
7
00:00:40,392 --> 00:00:45,527
We'll do that by adding a term to this equation.
8
00:00:49,142 --> 00:00:56,402
I've modified this equation by adding this term 1-P over A.
9
00:00:56,402 --> 00:01:04,838
The quantity A is known as an annihilation population; if the rabbits ever reach this population A, the next year there will be no rabbits.
10
00:01:05,018 --> 00:01:13,993
So this must be some level such that the rabbits will eat all their food, or fight with each other so badly that the next year there will be no rabbits.
11
00:01:14,374 --> 00:01:19,499
A is an annihilation population. You could also think of it as an "apocalypse" population.
12
00:01:24,622 --> 00:01:26,051
It's "we ever reach this population, the next year there will be no more rabbits."
13
00:01:26,387 --> 00:01:30,890
So let me show you that statement algebraiclally.
14
00:01:31,122 --> 00:01:35,215
If the population equals A, all the rabbits die. Let's see how that would work.
15
00:01:38,911 --> 00:01:41,517
The question is, "what's F of A?"
16
00:01:41,517 --> 00:01:48,248
Remember that this function tells you the population next year, if you know the population is P the current year.
17
00:01:48,248 --> 00:01:52,639
So if the population is A, f of A would give you the population next year.
18
00:01:52,639 --> 00:01:55,265
Lets plug in and see what happens.
19
00:01:55,265 --> 00:02:07,668
So I have r, I'm plugging in A for P
20
00:02:07,668 --> 00:02:15,241
A over A is 1,
21
00:02:15,241 --> 00:02:18,160
1 minus 1 is zero,
22
00:02:23,824 --> 00:02:25,701
and rA times zero is zero.
23
00:02:25,701 --> 00:02:31,600
Indeed, if the population reaches A, there will be no rabbits next year.
24
00:02:31,600 --> 00:02:37,200
So A is the annihilation population, or the apocalypse population.
25
00:02:37,200 --> 00:02:44,934
There is another property of this equation that I want to mention, that I think make sense;
26
00:02:44,934 --> 00:02:54,058
and that is if the populaiton is small, it is very close to the original model we studied.
27
00:02:54,058 --> 00:03:00,262
Let me write that out.
28
00:03:00,262 --> 00:03:08,200
For small populations, that are very far away from the apocalypse or the annihilation, we should see the rapid growth that we saw earlier.
29
00:03:08,200 --> 00:03:15,280
It's only when P gets to be large that the rabbits start to run out of food, or are competing with each other for space or something,
30
00:03:15,280 --> 00:03:18,724
that the population growth might start to slow down.
31
00:03:18,724 --> 00:03:31,000
First I should say what I mean by small. By small I mean that P is much less than A...
32
00:03:40,968 --> 00:03:45,151
and if that's the case, P over A is approximately zero.
33
00:03:45,151 --> 00:03:52,763
So maybe A is ten thousand, ten thousand rabbits is the apocalypse number, and P might be ten or twenty.
34
00:03:52,763 --> 00:03:58,811
Ten or twenty over ten thousand is a very small number. It's close to zero.
35
00:03:58,811 --> 00:04:16,344
When that's the case, P over A is close to zero. One minus something close to zero is close to one, this will be just a little bit less than one.
36
00:04:16,344 --> 00:04:32,691
So this term in parentheses here is around one. This equation just becomes rP.
37
00:04:32,691 --> 00:04:40,102
For small populations, this logistic equation--this is a logistic equation
38
00:04:40,102 --> 00:04:45,501
the population will grow rapidly like the rabbit example from the previous video.
39
00:04:45,907 --> 00:04:51,036
But then, once the population gets large, this term starts to matter and the population growth will slow down
40
00:04:51,915 --> 00:05:00,369
and there's some absolute upper limit and annihilation or apocalypse number at which the population will completely crash.
41
00:05:00,982 --> 00:05:13,920
So these two properties, I hope, seem reasonable for a simple model whose only goal is to have some limit to the rabbit growth.
42
00:05:15,655 --> 00:05:21,728
Next, I'll plot this equation, I'll plot this function of P and we'll interpret it graphically.
43
00:05:27,040 --> 00:05:34,294
The logistic equation is in this form: f of P is rP times 1 minus P over A.
44
00:05:35,212 --> 00:05:40,350
And this equation, when iterated, would describe the growth of a population.
45
00:05:41,176 --> 00:05:49,432
The idea is down here, I'm going to plot this function. On the horizontal axis, I'll have Pn, the population this year.
46
00:05:50,216 --> 00:05:55,718
And on the vertical axis, this function tells me what the population will be next year.
47
00:05:56,013 --> 00:06:01,770
So this--next year's population--you could call it F of P, or Pn plus one.
48
00:06:01,810 --> 00:06:15,671
So let's make a quick sketch of this function, and it turns out that it looks like this: It's an upside-down parabola.
49
00:06:15,671 --> 00:06:20,642
Obviously, just a rough sketch, but let's see what it tells us.
50
00:06:20,642 --> 00:06:28,809
If the population is small, then we have population growth. These units are arbitrary...
51
00:06:28,809 --> 00:06:34,706
If I'm at one unit here, then the population the next year would be two.
52
00:06:34,706 --> 00:06:41,898
We're about doubling...if I'm at two, then the population the next year might be very roughly four.
53
00:06:41,898 --> 00:06:50,856
So here, for a small population, we have pretty rapid growth.
54
00:06:50,856 --> 00:06:58,593
Here, if we're at the annihilation or apocalypse population, we have a very large population this year,
55
00:06:58,593 --> 00:07:03,680
The purple curve (the function) goes through zero, so that means that there will be zero rabbits the next year,
56
00:07:03,680 --> 00:07:07,160
Hence, annihilation or apocalypse.
57
00:07:07,160 --> 00:07:10,360
And if we're close to this value, there will be close to zero the next time.
58
00:07:10,360 --> 00:07:17,899
If I'm here, I'm very close to the maximum value--the apocalypse value--then what's my population next year?
59
00:07:17,899 --> 00:07:21,673
I'd go up here; the height of the graph tells me it will be quite small.
60
00:07:21,673 --> 00:07:29,560
So, as we approach the apocalypse value, the population is going to get smaller.
61
00:07:29,560 --> 00:07:36,676
In any event, the logistic equation: Here it is, it looks like this; we'll see graphs of this again.
62
00:07:36,676 --> 00:07:46,217
It is a simple model designed to capture population growth where there is some limiting factor to the population.
63
00:07:46,217 --> 00:07:50,669
It can't grow forever, there is some maximum value that--if it ever reaches--
64
00:07:50,669 --> 00:07:55,262
you suddenly lose all the population.
65
00:07:57,680 --> 00:08:03,711
I'll complete the derivation of the logistic equation by simplifying this equation a little bit,
66
00:08:03,711 --> 00:08:09,033
and putting it in a slightly different and more standard--and I think more general--form.
67
00:08:09,033 --> 00:08:18,665
Here is the logistic equation, rP times 1 minus P over A; A is the annihilation population, and r is a growth parameter,
68
00:08:18,852 --> 00:08:24,995
and this tells me the population next year, if I know the population this year.
69
00:08:25,358 --> 00:08:33,238
Let me write that in a slightly different way: Pn plus 1, the population next year,
70
00:08:33,500 --> 00:08:42,789
is r times Pn times 1 minus Pn over A.
71
00:08:43,063 --> 00:08:58,730
To simplify things a little bit, I'm going to divide both sides of the equation by A.
72
00:08:58,908 --> 00:09:01,320
I'm going to divide this by A,
73
00:09:01,320 --> 00:09:05,100
and I'm going to divide this by A.
74
00:09:05,520 --> 00:09:10,886
Mathematically, that's a legal move; I'm allowed to divide both sides of the equation by A,
75
00:09:10,886 --> 00:09:14,043
as long as A isn't zero, which it won't be.
76
00:09:14,043 --> 00:09:16,467
That will preserve the inequality.
77
00:09:16,467 --> 00:09:24,000
And then, notice I've got a P over A, a P over A, and a P over A and that will let me simplify things a little bit.
78
00:09:24,000 --> 00:09:29,043
So I'm going to define a new variable x, as follows.
79
00:09:35,553 --> 00:09:45,001
I'll define this new variable x as P divided by A. It's the population, but expressed as a fraction of the annihilation population.
80
00:09:45,001 --> 00:09:50,983
So, if x equals 0.5, that means we're halfway to the annihilation or apocalypse value;
81
00:09:50,983 --> 00:09:53,920
we're halfway to the maximum possible number.
82
00:09:53,920 --> 00:10:00,014
If x is point 8, then we're 80% of the way; if x is point one, we're just 10% of A.
83
00:10:00,014 --> 00:10:05,776
x, then, is a number that's always between zero and one.
84
00:10:05,776 --> 00:10:10,066
I can use this x to write this equation in a simpler form.
85
00:10:10,066 --> 00:10:17,202
P over A is x,
86
00:10:17,202 --> 00:10:28,122
P over A is x, P over A is x,
87
00:10:28,122 --> 00:10:32,854
So this tells me that next year's population is equal to rx times one minus x,
88
00:10:32,854 --> 00:10:39,218
where x is the population expressed as a fraction of this maximum possible value.
89
00:10:39,218 --> 00:10:44,035
Let me write this as a fuction, as well.
90
00:10:44,035 --> 00:10:52,269
The function we'll be working with that we'll be iterating a lot in the next couple of sub-units,
91
00:10:52,269 --> 00:10:58,054
is just this: F of x is Rx times one minus x.
92
00:10:58,054 --> 00:11:08,605
This is the logistic equation in the standard form that we'll work with. It's important enough that I'll put a red box around it.
93
00:11:08,605 --> 00:11:25,081
Just for completeness, r is a growth rate parameter, so r is something that will vary...will change,
94
00:11:25,081 --> 00:11:28,840
and we'll see how the behavior of this equation changes.
95
00:11:28,840 --> 00:11:43,310
Lastly, I can expand or multiply out the right-hand side of this, and I would get this:
96
00:11:43,310 --> 00:11:53,427
rx minus rx squared. So the logistic equation is just a second order polynomial, it's a parabola; it's a very simple function.
97
00:11:53,427 --> 00:12:00,977
You've studied parabolas in high school for sure, it's not an exotic or complicated function at all.
98
00:12:00,977 --> 00:12:10,147
In the next several subunits, we'll start iterating this function, and we'll see what the properties of the function are.
99
00:12:10,147 --> 00:12:16,554
I'll end this lecture with just a quick example, before you try one on your own.
100
00:12:18,737 --> 00:12:25,350
Let me do a quick example, just do review the idea of iterating the function.
101
00:12:25,350 --> 00:12:36,014
I'll iterate the logistic equation, and I'll let r equal 1.5. The function I'll be working with is f of x is 1.5x times one minus x.
102
00:12:36,014 --> 00:12:46,280
I need to choose a seed, so I'll see what happens if x equals zero point two.
103
00:12:46,280 --> 00:13:04,274
So the first iterate is obtained by applying the function to the seed, so that's 1.5 times 0.2 times one minus 0.2.
104
00:13:04,274 --> 00:13:10,173
So I'll do that on a calculator, let's see here...
105
00:13:14,023 --> 00:13:43,579
I get 0.24. OK, so then, the next iterate is F applied to 0.24, which is 1.5, times 0.24 times one minus 0.24.
106
00:13:43,579 --> 00:13:54,496
Let's do that on the calculator...
107
00:13:54,496 --> 00:14:11,424
I get 0.2736. We can keep going and going, and get the next iterate by applying the function again and again to the seed
108
00:14:11,424 --> 00:14:15,051
and then we can ask what its long term behavior is.
109
00:14:15,051 --> 00:14:17,760
We'll do that in the next sub-unit.
110
00:14:17,760 --> 00:14:27,459
Before you do so, I suggest you do the quiz that follows this lecture just to make sure you see how this goes.